Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6408564 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #34364 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60649 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1759 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,8,15,30)(4,18)(5,11,23,37)(6,13)(7,10)(9,22)(12,25)(14,31,39,56)(16,29)(17,38,26,28)(19,27,44,52)(20,43,46,57)(21,32)(24,35,49,36)(33,58,59,47)(34,48)(40,55)(41,54)(42,60)(45,51)(50,53)(61,64)(62,63);(1,3)(2,8)(4,19)(5,16)(6,15)(7,26)(9,31)(10,28)(11,29)(12,38)(13,30)(14,22)(17,25)(18,27)(20,45)(21,44)(23,41)(24,50)(32,52)(33,55)(34,56)(35,53)(36,60)(37,54)(39,48)(40,47)(42,49)(43,51)(46,62)(57,63)(58,61)(59,64);(1,4,7,22,6,21,25,48)(2,9,12,18,13,34,10,32)(3,14,17,19,15,39,26,44)(5,20,24,47,23,46,49,58)(8,27,28,31,30,52,38,56)(11,33,36,43,37,59,35,57)(16,40,42,45,41,61,50,62)(29,51,53,55,54,63,60,64);(2,10,13,12)(3,15)(8,28,30,38)(9,32,34,18)(11,35,37,36)(14,39)(16,41)(17,26)(19,44)(27,31,52,56)(29,53,54,60)(33,57,59,43)(40,61)(42,50)(45,62)(51,55,63,64);(1,5)(2,11)(3,16)(4,20)(6,23)(7,24)(8,29)(9,33)(10,35)(12,36)(13,37)(14,40)(15,41)(17,42)(18,43)(19,45)(21,46)(22,47)(25,49)(26,50)(27,51)(28,53)(30,54)(31,55)(32,57)(34,59)(38,60)(39,61)(44,62)(48,58)(52,63)(56,64)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,c^b=cg,d^b=df,g^b=gi,h^b=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6315763);